Seminari de Geometria Algebraica de Barcelona
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 Seminari de Geometria Algebraica 2022/2023 Conferenciant Títol Data i hora Kiran Kedlaya University of California San Diego Contact: francesc.fite at gmail.com The relative class number one problem for function fields Building on my lecture from ANTS-XV, we classify extensions of function fields (of curves over finite fields) with relative class number 1. Many of the ingredients come from the study of the maximum number of points on a curve over a finite field, such as the function field analogue of Weil's explicit formulas (a/k/a the "linear programming method"). Additional tools include the classification of abelian varieties of order 1 and the geometry of moduli spaces of curves of genus up to 7. Divendres 9 de setembre, 15h10, Aula T1, FMI-UB. Alexandre Turull University of Florida Contact:ignasi.mundet at ub.edu Representations of finite groups over small fields Let $$G$$ be a finite group and let $$F/K$$ be a field extension. A representation of $$G$$ over $$F$$ is a group homomorphism from $$G$$ to $$GL_n(F)$$, the general linear group of all automorphisms of a vector space of dimension $$n$$ over $$F$$. What representations of $$G$$ over $$F$$ arise from representations of $$G$$ over $$K$$ by simply composing with a natural homomorphism $$GL_n(K) \to GL_n(F)$$? We will discuss how to answer such questions in many cases. Along the way we will discuss $$p$$-adic numbers, Brauer groups, Brauer characters, and how to calculate certain invariants associated with irreducible representations of finite groups. These invariants may be different for Galois conjugate representations, but are nevertheless uniquely defined in the same sense that Brauer characters are uniquely defined. Divendres 30 de setembre, 15h10, Aula B1, FMI-UB. Barbara Fantechi SISSA Contact:marchesi at ub.edu MINI-CURS: Abelian covers and their moduli spaces One of the oldest methods to construct new smooth projective varieties from those we already know is to take cyclic covers branched over a smooth divisor. We will introduce this "bottom up" approach and apply to study moduli problems. In the first lecture, we will review this construction, and outline how to generalize it to abelian covers (i.e., finite Galois covers with abelian Galois group), following Pardini's work. In particular, we will show how geometric information about the cover can be described in terms of the base of the cover and the "branching data" (line bundles and divisors). In the second lecture we will introduce some key notions in infinitesimal deformation theory, and then apply them to study infinitesimal deformations of abelian covers, following a joint work with Pardini. In the third lecture we will briefly recall what a moduli space for varieties with ample canonical divisor is, and apply what we learned to construct examples of such moduli spaces. In particular we will discuss Vakil's lovely "Murphy's Law in Algebraic Geometry" paper, as well as work in progress with Pardini on moduli of surfaces of general type fibered in hyperelliptic curves. Dimarts 11, 18 i 25 d'octubre, 15h00, Aula IMUB, IMUB-UB. Irene Spelta Universitat de Barcelona Contact:irene.spelta01 at universitadipavia.it Prym maps and generic Torelli theorems: the case of plane quintics The talk deals with Prym varieties and Prym maps. Prym varieties are polarized abelian varieties associated with finite morphisms between smooth curves. Prym maps are accordingly defined as maps from the moduli space of coverings to the moduli spaces of polarized abelian varieties. Once recalled the classical generic Torelli theorem for the Prym map of étale double coverings, we will move to the more recent results on the ramified Prym map $$P_{g,r}$$ associated with ramified double coverings. For most of the values of $$(g,r)$$ a generic Torelli theorem holds and, furthermore, a global Torelli theorem holds when $$r$$ is greater (or equal to) 6. At the same time, it is known that $$P_{g,2}$$ and $$P_{g,4}$$ have positive dimensional fibres when restricted to the locus of coverings of hyperelliptic curves. But this is not a characterization: the study of the differential $$dP_{g,r}$$ shows that there are also other configurations to be considered. We will focus on the case of degree 2 coverings of plane quintics ramified in 2 points. We will show that the restriction of $$P_{g,r}$$ here is generically injective. This is joint work with J.C. Naranjo. Divendres 21 d'octubre, 15h10, Aula B1, FMI-UB. Martí Salat Moltó Universitat de Barcelona Contact:marti.salat at ub.edu Equivariant sheaves and vector bundles on toric varieties In this talk, we consider the theory of equivariant sheaves on a toric variety described via (multi)filtrations of a vector space, as introduced by Klyachko in the 90s. Starting with torsion-free equivariant sheaves, we will apply this theory to give a description of monomial ideals, generalizing the classical staircase diagrams and suited to non-standard gradings of a polynomial ring. Afterwards, we will focus on equivariant reflexive sheaves and vector bundles. We introduce a family of lattice polytopes encoding their global sections. In the case of Picard rank 2 smooth projective toric varieties, this description allows us to compute explicitly the Hilbert polynomial of an equivariant reflexive sheaf. Divendres 28 d'octubre, 15h10, Aula B1, FMI-UB. Luis Núñez-Betancourt CIMAT-CRM Contact:josep.alvarez at upc.es Nash blowup in prime characteristic The Nash blowup is a natural modification of algebraic varieties that replace singular points by limits of certain vector spaces associated to the variety at non-singular points. For several decades it has been studied whether it is possible to resolve singularities of algebraic varieties by iterating Nash blowups. This problem has mostly been treated in characteristic zero due to an example given by Nobile. In this talk, we will discuss a new approach in prime characteristic using differential operators, and an application to resolution of singularities of toric varieties. Divendres 4 de novembre, 15h10, Aula B1, FMI-UB. Raheleh Jafari Kharazmi University Contact:szarzuela at ub.edu On the Gorenstein locus of simplicial affine semigroup rings The Gorenstein locus of simplicial affine semigroup rings is studied by an analysis of Cohen-Macaulay type of homogeneous localizations at monomial prime ideals. In particular, we discuss a geometrical characterization of the semigroup rings that are Gorenstein on the punctured spectrum. Divendres 18 de novembre, 15h10, Aula B1, FMI-UB. Lothar Göettsche ICTP - International Centre for Theoretical Physics Contact:marchesi at ub.edu (Refined) Verlinde and Segre formulas for Hilbert schemes of points This is joint work with Anton Mellit. Segre and Verlinde numbers of Hilbert schemes of points have been studied for a long time. The Segre numbers are evaluations of top Chern and Segre classes of so-called tautological bundles on Hilbert schemes of points. The Verlinde numbers are the holomorphic Euler characteristics of line bundles on these Hilbert schemes. We give the generating functions for the Segre and Verlinde numbers of Hilbert schemes of points. The formula is proven for surfaces with $$K_S^2=0$$, and conjectured in general. Without restriction on $$K_S^2$$ we prove the conjectured Verlinde-Segre correspondence relating Segre and Verlinde numbers of Hilbert schemes. Finally we find a generating function for finer invariants, which specialize to both the Segre and Verlinde numbers, giving some kind of explanation of the Verlinde-Segre correspondence. Divendres 25 de novembre, 15h10, Aula B1, FMI-UB. Giuseppe Ancona Université de Strasbourg Contact:ffite at ub.edu Quadratic forms arising from geometry The cup product on topological manifolds or the intersection product on algebraic varieties induce quadratic forms which turn out to be a fine invariant of these geometric objects. We will discuss some old theorems on the signature of these quadratic forms and some applications both of geometric and arithmetic origins. Finally we will study an old conjecture of Grothendieck about those signatures and explain some new evidences. Dimarts 29 de novembre, 17h00, Aula B7, FMI-UB. Daniel Macias Castillo UAM-ICMAT Contact:francesc at mat.uab.edu Towards a refined class number formula for Drinfeld modules This is joint work with María Inés de Frutos Fernández and Daniel Martínez Marqués. In 2012, Taelman proved an analogue of the Analytic Class Number Formula, for the Goss L-functions that are associated to Drinfeld modules. He also explicitly stated that "it should be possible to formulate and prove an equivariant version" of this formula. We report on work in progress motivated by Taelman's statement. We formulate a precise equivariant, or `refined', generalisation of Taelman's formula, and we provide very strong evidence to support it. In the general case, we reduce the validity of our refined class number formula to a purely algebraic problem in the K-theory of positive characteristic group rings. As a concrete consequence of our general approach, we also prove (unconditionally) a natural non-abelian generalisation of a Theorem of Anglès and Taelman, regarding the Galois structure of Taelman's class group. Divendres 2 de desembre, 15h10, Aula B1, FMI-UB. Barbara Fantechi SISSA Contact:marchesi at ub.edu TBA Divendres 16 de desembre, 15h10, Aula B1, FMI-UB. Ngo Viet Trung Vietnam Academy of Science and Technology Contact:szarzuela at ub.edu TBA Divendres 20 de gener, 15h10, FMI-UB. Carles Checa National Kapodistrian University of Athens Contact:marchesi at ub.edu TBA Divendres 3 de febrer, 15h10, FMI-UB.

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